Abstract
In this paper, velocity and attenuation of ultrasonic S-wave in a water-saturated rock are used for calculating shear modulus and matrix permeability of the rock, via a model improved from Biot theory. The model requires two inputs, i.e., the dry velocity of S-wave and the average distance of aperture representing pores, to yield phase velocity and the quality factor as functions of frequency. By fitting the predicted velocity and quality factor against the ultrasonically measured counterparts, the dry velocity of S-wave and the average distance of aperture are ascertained, which in turn yield shear modulus and matrix permeability, respectively. The modeling results on D’Euville limestone from France show that the specimen has shear modulus of 11.35 and 11.55 GPa (under differential pressures of 3 and 5 MPa, respectively) and matrix permeability of 0.0486 Darcy (under both differential pressures). The matrix permeability appears to be approximately one half of Darcy permeability.
1. Introduction
In elastic mechanics, shear modulus refers to that at very low frequencies, or the quasi-static state. Early rock physicists tended to a consensus that at very low frequencies, shear modulus of a fluid-saturated rock is unaffected by the fluid [1–5]. In other words, shear modulus is the same between the saturated rock and the dry rock (also called skeleton). The reason is that fluid has vanishing shear modulus and therefore does not contribute to the shear modulus of a rock.
For ultrasonic P-wave, laboratory observations showed that the velocity of fluid saturated rocks is high and the attenuation is large [6–8], which cannot be interpreted by Biot theory [2, 3]. Hence, squirt models [9–14] were proposed; the idea was that P-wave induces a strong squirt between the narrow gaps at contact of grains and the main pore space, which significantly increases the wave velocity and attenuates the mechanical energy. Meanwhile, the squirt models [9–14] hypothesized that S-wave had the same squirt mechanism as P-wave does.
However, S-wave is distinctly different from P-wave in that S-wave only involves shear stress/strain, or angular change [1–5, 15, 16], and therefore, extending the squirt mechanism of P-wave to S-wave is incorrect. In this regard, this paper agrees with the early rock physicists on the consensus that S-wave will not cause squirt.
According to the Darcy law [5], Darcy permeability () is a parameter used along with fluid viscosity and velocity difference (between fluid and solid) to quantify the viscous stress between them at very low frequencies. At very low frequencies, the profile of fluid flow in pores is parabolic, such that a single value of is well applicable. However, at high frequencies, the profile of fluid flow is frequency dependent [17], and a single value of alone is insufficient to quantify the viscous stress as a function of frequency. Biot [2] used and the coupling factor () to quantify the viscous stress between the two phases, but was incapable of well predicting velocity and attenuation of ultrasonic S-wave measured on fluid-saturated rocks [15, 16]. Biot [3] used dynamic fluid viscosity (a function of frequency) plus to quantify the viscous stress, which was redundant because his dynamic fluid viscosity alone is sufficient to quantify the viscous stress between solid and fluid.
More recently, based on the Stokes second problem [17, 18], Li [15, 16] improved the S-wave theory in Biot [2, 3] to a non-squirt model which well predicted velocity and attenuation of ultrasonic S-wave in Berea sandstone and Boise sandstone. Recall that is a parameter used along with fluid viscosity and velocity difference (between fluid and solid) to quantify the viscous stress between them at very low frequencies. Similar to that, the S-wave permeability [15, 16] is a (frequency-dependent) quantity used along with fluid viscosity and the velocity difference to quantify the viscous stress when a monochromatic S-wave propagates through the rocks. The S-wave permeability [15, 16] was equivalent to dynamic fluid viscosity in Biot [3], except that the redundant was deleted because the frequency-dependent permeability alone is sufficient to quantify the viscous stress [17]. In [15, 16], it was assumed to be a real number for simplicity. Notably, at high frequencies, the viscous stress between solid and fluid may not be synchronous with the relative velocity between them [3], meaning that the permeability at a high frequency is likely to be a complex number.
When the S-wave propagates through a fluid-saturated rock, it has to scan both matrix pores and large fissures. As matrix pores host the majority of water, the propagation of S-wave is largely controlled by the matrix pores rather than by the fissures. As such, the S-wave permeability at the low-frequency limit is referred to as matrix permeability.
In this study, the non-synchronization of the viscous stress with the relative velocity between the two phases is simulated by extending the real-number S-wave permeability to a complex number. Substituting the permeability into the non-squirt model of S-wave [15, 16] yields predictions of velocity and attenuation as functions of frequency (Section 2). In Section 3, a well-studied carbonate rock, D’Euville limestone, is employed as an application example; the core sample is used to validate the model because of the homogeneity of the limestone. The velocity and quality factor of ultrasonic S-wave measured on the water-saturated sample [19] are used via the model to obtain shear modulus and matrix permeability. The final is discussion of the limestone from France, in comparison with other limestones from the central United States.
2. S-Wave Theory
2.1. Permeability as a Function of Frequency
Similar to ([3], Section 2), the pores of a rock matrix can be treated as a 2D planar aperture. According to equation (2.12) in Biot [2], the shear strain of solid in a fluid-saturated rock is associated with the shear stress onto the solid alone, irrelevant to fluid pressure. For this reason, fluid pressure is set to vanish for study of S-wave. For S-wave, the fluid motion in the aperture is a boundary-value problem in which active solid at the pore wall pulls passive fluid (which is precisely the mechanism of the Stokes second problem [17]). Different from Biot [3] in which the fluid viscosity was replaced by dynamic fluid viscosity to account for the effect caused by the oscillating fluid flow at high frequencies, Darcy permeability is replaced with the S-wave permeability, while the fluid viscosity is kept unchanged.
A notable quantity is permeability angle () defined as where is dimensionless angular frequency defined as follows: where is the angular frequency, b is the aperture distance and is the fluid kinematic viscosity ( is the fluid dynamic viscosity and is the fluid density).
The S-wave permeability turns out to be ([3], Section 2):
Recall that for the planar aperture, matrix permeability is (equation (5.10.10) in [20]). Equation (3) can be rewritten as
Rock matrix refers to the porous rock free of fissures. Compared with the pores, fissures allow for a much more rapid flow due to their much higher permeability. Flow in the matrix is usually very slow due to the narrow throats between pores. should be close to , but there are no assurances that they equal to each other.
2.2. Phase Velocity and the Quality Factor
Follow the S-wave wavenumber () equation [15, 16]. As the viscous stress between solid and fluid has been well considered, there is no need to include the coupling density () into the momentum equations [2, 3] any longer, which is the difference of our model from Biot [3]: where is the wave velocity of the skeleton ( is the shear modulus of the skeleton and is the density of the skeleton) and denotes the porosity.
The S-wave permeability, , is a complex number, i.e., . Substituting this expression into Equation (5) yields where and are two buffer (real) numbers.
Phase velocity is denoted as , while the quality factor is denoted as . Solving the square root of complex number yields [21]: where .
Our model has two unknowns, i.e., aperture distance () and S-wave velocity of dry rock (). On the other hand, the ultrasonic measurements yielded two quantities, i.e., the ultrasonic velocity () and quality factor (). Using the two measured quantities into the model shall ascertain and ; the problem is neither under-determinate nor over-determinate. Finally, and can be converted to S-wave permeability () and shear modulus (G), respectively.
3. Application to D’Euville Limestone
The D’Euville limestone is a carbonate rock from Euville, France. Forming in the Upper Jurassic period, the limestone is oolitic with angular grains. It is composed of approximately 98% calcite. It consists of crinoids with a diameter of 500-2000 μm and can be characterized as a grainstone [22]. Its mean porosity ranges from 0.15 to 0.20, and Darcy permeability is 0.0837 Darcy under water-vapor permeability test [23].
Lucet and Zinszner [19] conducted ultrasonic and sonic measurements on a water-saturated sample of D’Euville limestone, with the use of piezoceramic transducers and a resonant bar, respectively. Unfortunately, they did not measure on the dry sample. Mavko and Jizba [10] and Dvorkin et al. [11] attempted to simulate the measured velocity and quality factor, based on the assumption that the sonic velocity dispersion on the saturated sample [19] was so small that the sonic velocity measured on the saturated sample can be converted to that of the dry sample (see Table 1 in [11]). The implicit assumption was that the shear modulus is the same between the saturated sample at sonic frequency and the dry sample. However, the resonant-bar method was not quite accurate, probably because the containing apparatus (that maintained water saturation and differential pressure) would resonate along with the bar itself, which caused errors in the measurement results. For this reason, the sonic data is excluded from this paper, and we only use the ultrasonic data.
Tables 1 and 2 list the parameters of D’Euville limestone, for differential pressures of 3 and 5 MPa, respectively. At frequency of 500 kHz, the measured velocity () was 2209 and 2229 m/s, for differential pressures of 3 and 5 MPa, respectively [11, 19]. Under both differential pressures, the quality factor () was measured to be 7.
In the modeling, we change the aperture distance () and the dry velocity of S-wave (), to see how the predicted velocity () and quality factor () fit the measured counterparts. For differential pressure of 3 MPa, the model with and predicts of 2210 m/s and of 31 at 0.5 MHz (see Figures 1 and 2). The modeled S-wave permeability is depicted with frequency in Figure 3. For differential pressure of 5 MPa, the model with and predicts of 2229 m/s and of 31 at 0.5 MHz (see Figures 4 and 5). The modeled is depicted with the frequency in Figure 6.
Under both differential pressures, the ultrasonic velocity () is the same between the model and measurement. However, the predicted (31) is much higher than the measured (7). Ideally, induced by viscous flow alone should be used for modeling. However, Lucet and Zinszner [19] only measured the total attenuation that consists of the viscous-flow-induced attenuation and the skeleton/dry attenuation. With the consideration that the total attenuation is invariably larger than the viscous-flow-induced attenuation, the fitting between model and measurement in Figures 2 and 5 is actually better than at the first glance.
Often, ultrasonic velocity () is accurately measured, and our fitting of is strict. For ultrasonic (which is more difficult to measure), the fitting is required to be optimal among all combinations of and , or equivalently and . As a reference, we set for simulation. As plotted in Figure 7, the fitting of is pretty well. However, the fitting of is worse (Figure 8 versus Figure 2). From Figure 9, the matrix permeability reads Darcy.
4. Discussion
The modeling results on D’Euville limestone show that the specimen [19] has shear modulus of 11.35 GPa (under differential pressure of 3 MPa) and 11.55 GPa (under differential pressure of 5 MPa). According to [19] and Table 1 in [24], the shear modulus was 9.42 and 10.2 GPa, for differential pressures of 3 and 5 MPa, respectively. Eide [25] measured the shear modulus of two samples of D’Euville limestone with porosity of 0.165 and 0.169, to be 10.8-11.2 GPa, which is slightly lower than our shear modulus (11.35-11.55 GPa). The lower shear modulus of their samples than the specimen in [19] is consistent with the samples in [25] having more fissures and much higher Darcy permeability than the specimen in [19].
At the low limit of frequency, the right hand side of equation (4) will tend to unitary, meaning a matrix permeability of 0.0486 Darcy (please see the low-frequency end of Figures 3 and 6). of the D’Euville limestone specimen [19] is 0.1 Darcy (Table 1 in [24]). Our matrix permeability appears to be approximately one half of the Darcy permeability. According to water-vapor permeability test [23], D’Euville limestone had of 0.0837 Darcy.
According to (Table C1 in [26]), Indiana limestone (also called as Bedford limestone because of its outcrop in Bedford, Indiana, the USA) has a porosity of 0.13 (lower than D’Euville limestone in porosity) and a shear modulus of 12.1 GPa (higher than ours). This suggests that Indiana limestone [26] is relatively intact (fissures-free). The higher of Indiana limestone than D’Euville limestone is consistent with the former limestone forming during the Mississippian age while the latter during the Upper Jurassic period.
Madison limestone in the central USA (from the Rocky Mountain to the Great Plains) is also a thick sequence of carbonate rocks during the Mississippian age. The intact sample of Madison limestone has a mean permeability of 0.0158 Darcy [27]. The lower permeability of Madison limestone (than D’Euville limestone) suggests that it is more fissures-free than D’Euville limestone. Nevertheless, limestones have a very wide range of permeability, to a large extent depending on fissures. Fissures, if present in a limestone, act as preferential pathways for seepage and enhance much higher than [28]. The permeability inverted via our model represents the matrix pores which is comparable to probably because not many fissures were observed in the D’Euville limestone specimen.
At last, porosity in rock physics often refers to open porosity or effective porosity. The high porosity of D’Euville limestone (0.18) [19] also suggests its diagenesis under an environment of strong water dynamics (a static water environment would result in a very low porosity), as well as its intactness (the lack of many fissures). The presence of many fissures on a limestone is often associated with a low porosity, because fissures are produced by strong tectonic stress which on the other hand decreases the original porosity by compression and recrystallization [29].
5. Conclusions
(1)This study introduces a new method which utilizes acoustic data, i.e., velocity and quality factor of ultrasonic S-wave measured on a fluid saturated rock, to get its shear modulus and matrix permeability. The method is reliable in that the mathematical problem is well-determinate, i.e., two measured quantities to determinate two unknowns(2)The modeling on D’Euville limestone yields the results that the specimen has shear modulus of 11.35 and 11.55 GPa (under differential pressures of 3 and 5 MPa, respectively). Under both differential pressures, the resulting matrix permeability appears to be 0.0486 Darcy, which is approximately one half of Darcy permeability measured from seepage experiments
Nomenclature
| : | Aperture distance |
| : | S-wave velocity of skeleton () |
| : | Frequency |
| : | Shear modulus of skeleton |
| : | Wavenumber |
| : | The quality factor of S-wave |
| : | S-wave velocity |
| : | Darcy permeability |
| : | Matrix permeability |
| : | S-wave permeability |
| : | The real part of S-wave permeability |
| : | The imaginary part of S-wave permeability |
| : | Fluid dynamic viscosity |
| : | Fluid kinematic viscosity () |
| : | The total density |
| : | Fluid density |
| : | Skeleton density |
| : | Permeability angle |
| : | Rock porosity |
| : | Angular frequency |
| : | Dimensionless angular frequency. |
Data Availability
The data yielding from the model is available with doi:10.6084/m9.figshare.16802068 at https://figshare.com/s/520c4e651b2d88b4e3a9.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Acknowledgments
The study was sponsored by the National Natural Science Foundation of China under grant 42064006.